The field of the invention is nuclear magnetic resonance imaging (MRI) methods and systems. More particularly, the invention relates to the reconstruction of images from acquired MR data that samples k-space in a non-uniform manner.
When a substance such as human tissue is subjected to a uniform magnetic field (polarizing field B0), the individual magnetic moments of the excited nuclei in the tissue attempt to align with this polarizing field, but precess about it in random order at their characteristic Larmor frequency. If the substance, or tissue, is subjected to a magnetic field (excitation field B1) that is in the plane orthogonal to the main polarizing field (generally designated the x-y plane) and that is near the Larmor frequency, the net aligned moment, Mz, may be rotated, or “tipped”, into the x-y plane to produce a net transverse magnetic moment Mt. A signal emitted by the excited nuclei or “spins” may be received after the excitation signal Bt is terminated, and may be processed to form an image.
When utilizing these “MR” signals to produce images, magnetic field gradients (Gx, Gy and Gz) are employed. Typically, the region to be imaged is scanned by a sequence of measurement cycles in which these gradients vary according to the particular localization method being used. The resulting set of received MR signals are digitized and processed to reconstruct the image using one of many well known reconstruction techniques.
The measurement cycle used to acquire each MR signal is performed under the direction of a pulse sequence produced by a pulse sequencer. Clinically available MRI systems store a library of such pulse sequences that can be prescribed to meet the needs of many different clinical applications. Research MRI systems include a library of clinically proven pulse sequences and they also enable the development of new pulse sequences.
The MR signals acquired with an MRI system are signal samples of the subject of the examination in Fourier space, or what is often referred to in the art as “k-space”. Each MR measurement cycle, or pulse sequence, typically samples a portion of k-space along a sampling trajectory characteristic of that pulse sequence. Most pulse sequences sample k-space in a raster scan-like pattern sometimes referred to as a “spin-warp”, a “Fourier”, a “rectilinear” or a “Cartesian” scan. The spin-warp scan technique is discussed in an article entitled “Spin-Warp MR Imaging and Applications to Human Whole-Body Imaging” by W. A. Edelstein et al., Physics in Medicine and Biology, Vol. 25, pp. 751-756 (1980). It employs a variable amplitude phase encoding magnetic field gradient pulse prior to the acquisition of MR spin-echo signals to phase encode spatial information in the direction of this gradient. In a two-dimensional implementation (2DFT), for example, spatial information is encoded in one direction by applying a phase encoding gradient (Gy) along that direction, and then a spin-echo signal is acquired in the presence of a readout magnetic field gradient (Gx) in a direction orthogonal to the phase encoding direction. The readout gradient present during the spin-echo acquisition encodes spatial information in the orthogonal direction. In a typical 2DFT pulse sequence, the magnitude of the phase encoding gradient pulse Gy is incremented (ΔGy) in the sequence of measurement cycles, or “views” that are acquired during the scan to produce a set of k-space MR data from which an entire image can be reconstructed.
There are many other k-space sampling patterns used by MRI systems These include “radial”, or “projection reconstruction” scans in which k-space is sampled as a set of radial sampling trajectories extending from the center of k-space as described, for example, in U.S. Pat. No. 6,954,067. The pulse sequences for a radial scan are characterized by the lack of a phase encoding gradient and the presence of a readout gradient that changes direction from one pulse sequence view to the next. There are also many k-space sampling methods that are closely related to the radial scan and that sample along a curved k-space sampling trajectory rather than the straight line radial trajectory. Such pulse sequences are described, for example, in “Fast Three Dimensional Sodium Imaging”, MRM, 37:706-715, 1997 by F. E. Boada, et al. and in “Rapid 3° D. PC-MRA Using Spiral Projection Imaging”, Proc. Intl. Soc. Magn. Reson. Med. 13 (2005) by K. V. Koladia et al and “Spiral Projection Imaging: a new fast 3D trajectory”, Proc. Intl. Soc. Mag. Reson. Med. 13 (2005) by J. G. Pipe and Koladia.
An image is reconstructed from the acquired k-space data by transforming the k-space data set to an image space data set. There are many different methods for performing this task and the method used is often determined by the technique used to acquire the k-space data. With a Cartesian grid of k-space data that results from a 2D or 3D spin-warp acquisition, for example, the most common reconstruction method used is an inverse Fourier transformation (“2DFT” or “3DFT”) along each of the 2 or 3 axes of the data set. With a radial k-space data set and its variations, the most common reconstruction method includes “regridding” the k-space samples to create a Cartesian grid of k-space samples and then perform a 2DFT or 3DFT on the regridded k-space data set. In the alternative, a radial k-space data set can also be transformed to Radon space by performing a 1DFT of each radial projection view and then transforming the Radon space data set to image space by performing a filtered backprojection.
Reconstructing images from k-space samples acquired using other non-Cartesian trajectories can be more difficult. The acquired MR samples do not provide independent information about an imaged object over a selected field-of-view (FOV) and must therefore be weighted to account for non-uniform sampling density. This weighting process is referred to as density compensation and images reconstructed from unweighted MR data will include inaccuracies and will not properly represent an imaged object.
A traditional method of performing density compensation involves multiplying a density compensation function (DCF), which ideally accounts for the Jacobian in the Fourier transform integral relationship, with samples acquired using non-Cartesian acquisition trajectories. However, computing the Jacobian determinant requires the existence of an analytical transformation for the given trajectory to a uniformly unit-space coordinate system that may not be available for trajectories used in MRI. It is possible to numerically evaluate the DCF by defining area elements associated with each sample to mimic the Jacobian and by implementing goal-based optimization of the point-spread function (PSF) defined by the Fourier expansion of the DCF. However, numerical methods do not always adequately evaluate the DCF, as the optimization of the PSF relies on iterative refinement methods that do not guarantee convergence.
Other methods for performing density compensation employ linear systems to model the imaging or resampling problem. These methods often include the direct inversion of large and dense matrices, a computationally expensive process that is especially problematic when producing large MR images. Computational burden may be reduced by employing iterative linear system solvers and stopping criteria or by employing heuristic simplifications such as truncation or approximation. However, these simplification methods do not provide exact density compensation and generally lead to MR images having reduced accuracy.
It would therefore be desirable to provide a method for performing an accurate density compensation on MR data acquired using non-uniform acquisition trajectories. Such a method would reduce reconstruction errors when producing MR images and provide improved signal-to-noise ratio (SNR).